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A287650
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Number of doubly symmetric diagonal Latin squares of order 4n with the first row in ascending order.
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10
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OFFSET
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1,1
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COMMENTS
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A doubly symmetric square has symmetries in both the horizontal and vertical planes.
The plane symmetry requires one-to-one correspondence between the values of elements a[i,j] and a[N+1-i,j] in a vertical plane, and between the values of elements a[i,j] and a[i,N+1-j] in a horizontal plane for 1 <= i,j <= N. - Eduard I. Vatutin, Alexey D. Belyshev, Oct 09 2017
Belyshev (2017) proved that doubly symmetric diagonal Latin squares exist only for orders N == 0 (mod 4).
Every doubly symmetric diagonal Latin square also has central symmetry. The converse is not true in general. It follows that a(n) <= A293777(4n). - Eduard I. Vatutin, May 26 2021
a(n)/(A001147(n)*2^(n*(4*n-3))) is the number of 2n X 2n grids with two instances of each of 1..n on the main diagonal and in each row and column with the first row in nondescreasing order. - Andrew Howroyd, May 30 2021
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LINKS
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E. I. Vatutin, Special types of diagonal Latin squares, Cloud and distributed computing systems in electronic control conference, within the National supercomputing forum (NSCF - 2022). Pereslavl-Zalessky, 2023. pp. 9-18. (in Russian)
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FORMULA
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EXAMPLE
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Doubly symmetric diagonal Latin square example:
0 1 2 3 4 5 6 7
3 2 7 6 1 0 5 4
2 3 1 0 7 6 4 5
6 7 5 4 3 2 0 1
7 6 3 2 5 4 1 0
4 5 0 1 6 7 2 3
5 4 6 7 0 1 3 2
1 0 4 5 2 3 7 6
Reflection of all rows is equivalent to the exchange of elements 0 and 7, 1 and 6, 2 and 5, 3 and 4; hence, this square is horizontally symmetric. Reflection of all columns is equivalent to the exchange of elements 0 and 1, 2 and 4, 3 and 5, 6 and 7; hence, the square is also vertically symmetric.
a(2) = 4*3*1024 = 12288. The 4 base quarter square arrangements are:
1 1 2 2 1 1 2 2 1 1 2 2 1 1 2 2
1 2 1 2 1 2 2 1 2 2 1 1 2 2 1 1
2 1 2 1 2 2 1 1 1 1 2 2 2 2 1 1
2 2 1 1 2 1 1 2 2 2 1 1 1 1 2 2
(End)
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CROSSREFS
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KEYWORD
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nonn,more,hard
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AUTHOR
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EXTENSIONS
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Edited and a(3) from Alexey D. Belyshev added by Max Alekseyev, Aug 23 2018, Sep 07 2018
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STATUS
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approved
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